Do flying monks described in Buddhist texts contradict with physics?

Buddhism denies the existence of an objective universe. There is only the "packets" of sensation which if experienced un analysed contain no name nor form, and thus are chaos or void. We fudge-stitch our experiences together, assuming that things that are similar are the same. There never was any space to fly through nor up nor down. When you experience that purity, when you shake the other from your back, then at least it feels like you are flying, and other people who can see the fire in your eyes will feel like you are flying too.

Nietzsche is about the only Western philosopher who associates states of mind with freedom from gravity, which he expresses in the following way in Zarathustra

And when I saw my devil, I found him serious, thorough, profound, solemn: he was the spirit of gravity - through him all things fall. Not by wrath, but by laughter, do we slay. Come, let us slay the spirit of gravity! I learned to walk; since then have I let myself run. I learned to fly; since then I do not need pushing in order to move from a spot. Now am I light, now do I fly; now do I see myself under myself. Now there dances a God in me.-"

But perhaps Einstien himself experienced that rapture too. He explained "The happiest thought of my life" in the following way. The gravitational field has only a relative existence... Because for an observer freely falling from the roof of a house - at least in his immediate surroundings - there exists no gravitational field. https://physics.stackexchange.com/questions/13359/einsteins-happiest-thought

The Buddhist texts claim that many things are possible that many people think are not possible. As Suminda said, the claim is that once one properly develops the four bases of power, i.e., profound concentration founded on (1) desire & effort, (2) persistence, (3) intent, and (4) discrimination, then one will be capable of gaining 'supranormal powers,' as described in this passage from the Iddhipada-vibhanga Sutta:

When a monk has thus developed & pursued the four bases of power, he experiences manifold supranormal powers. Having been one he becomes many; having been many he becomes one. He appears. He vanishes. He goes unimpeded through walls, ramparts, & mountains as if through space. He dives in & out of the earth as if it were water. He walks on water without sinking as if it were dry land. Sitting crosslegged he flies through the air like a winged bird. With his hand he touches & strokes even the sun & moon, so mighty & powerful. He exercises influence with his body even as far as the Brahma worlds.

These appear, at least at first glance, to be literal, bodily powers; i.e., he bodily walks on water without sinking, although perhaps one could claim that the 'body' referred to in this passage refers to the 'mind-made body.' This is at least prima facie plausible since, in the Samaññaphala Sutta, the passage on the aformentioned powers occurs exactly as above but immediately follows a section on the 'mind-made body.' In either case, there are other powers mentioned in this sutta such as reading others' minds, recollecting past lives, and personally seeing how beings fare according to their kamma that contradict mainstream scientific views.

So, yes, there seems to be a contradiction between a worldview, perhaps a materialist scientific view, that holds that it is impossible to fly without technology and what is proclaimed in the suttas. Even if we interpret the suttas to be saying that one can physcially fly, I don't know that the Buddhist view conflicts with physics per se; to borrow from author Sam Harris on this point, it would be a discovery for physics if human flight based on concentration were shown to be possible.

Again, even if we allow that the suttas are talking about physical flight, built into the details is an explanation of why this contradiction should occur; namely, (1) the degree of concentration required for the development of these powers is not attained by the average person (hence, supranormal) and (2) Whatever Buddhist monks do attain these powers are forbidden by the monastic code from displaying these powers. Based on these factors, it would make sense that mainstream scientists or your average layperson would have no knowledge of these possibilities.

On the other hand, if the suttas are talking about the mind-made body (astral body), then although I think the details will still reveal a conflict with mainstream physics, it will not be as gross a contradiction. In any case, there certainly is a contradiction between the physicalist view that death is annihilation and the view that 'there is another world'--i.e., rebirth (cf. Apannaka sutta). So, ultimately Buddhism is at odds with mainstream views of physics.

"So the Buddhist texts must be stating something incorrectly since we now know that we can't fly without the help of technology. How could they fly then?"

What text are you referring to?

Also, what is meant by "flying"? Is it an actual body? Or, perhaps, just the mind?

"since we now know that we can't fly without the help of technology"

I don't think the above can be inferred from physics. Not much because I am a physicist, but because natural laws have a way of refusing to be subsumed by assertions like these. In other words, it seems to be another instance of the old induction trap. For example, seeing 100 black cats is enough to assert "all cats are black" without incurring in mistake?

The Buddha himself seemed to be acutely aware of this problem when he says he does not discern a beginning [of the world], instead of relapsing his reasoning and declaring "the world is eternal" (see Brahmajāla Sutta).

One of the main pieces of advice in the Buddha Dhamma is verification by trying it yourself (Ehipassiko) and to be open to scientific scrutiny (Sandiṭṭhiko). Hence you can develop the Jhanas and see for your self if this is possible or not. Self-verification and experience is more convincing than people arguing in-favor or against something like this.

Having said this, there is much research in these areas under Parapsychology though this might be view as pseudoscience.