Is mindfulness meditation banned in Eastern Orthodoxy?

Saint Vyacheslav Krasheninnikov was against deep breathing methods, needle therapy, massages, etc. Yogis levitate because demons lift them. Similarly, the antichrist flies quickly because demons carry him fast; deceived people say: "Christ is here; Christ is there!".

In the Eastern Orthodoxy there is a short prayer "Lord Jesus Christ, Son of God, have mercy on me, a sinner". It is an accepted practice to repeat this prayer many times, while focusing fully on the words and their meaning (instead of breath).

The primary point is to clear your mind from any thoughts besides this one, so the technique is similar. I think the primary difference between this prayer and "breathing techniques" is the fact that by praying you then can attribute all the improvements in the spiritual life to God. In contrast, if you focus on breathing, in your mind you eventually attribute any changes to yourself, so it does not guard you from pride. This is why it's considered imperfect, as prayer additionally has a chance to shield you from pride as you do not attribute the effects to yourself. I won't go as far as to say breathing meditation is banned, it's just that prayer technique is considered to be better as is solves the issue with pride.

Mindfulness is not "banned", to the contrary, it's the primary point of "ascesis" - to observe your thoughts carefully and reject thoughts that might harm you - to the point you no longer have them (which was achieved by just a handful of people).

Also my personal opinion: saying things like "if you do X, then the devil will enter your mind" sounds like and extremism and magical thinking to me - it's kind of similar to believing that by performing some ritual (like casting a spell) we can free ourselves from the influence of the devil, which I think is foreign to the Orthodoxy and Christianity overall. Eastern Orthodoxy believes that only God fully controls the devil and allows him to affect us only if it's for our benefit in the long run (so that means the devil does good against his will, thinking he is harming someone - which shows the ultimate cleverness of God), so it's not up to us to try to influence that, we should just focus on doing our best to maintain our moral values, lead a good life and be mindful.

Is mindfulness banned in the Eastern Orthodox church? No.

As is often the case, I don't think there is a concrete "Eastern Orthodox opinion" on this. You should consult with your priest, as his opinion will matter most, but here is the opinion of other Orthodox people.

Mindfulness itself is very important, as long as it's practiced in a Christian way. There is actually a Greek word that could be comparable to "mindfulness," "nepsis" (νῆψις). Steadying your thoughts is part of asceticism:

A mind that is left to its own devices will remain untrained. An untrained mind is impossible to control. It will remain jumping from one thought to another just like a butterfly in a field of flowers. It will quickly jump from one flower to the another in what seems like a random pattern. To develop mindfulness or watchfulness requires ascetic disciplines, such as prayer and fasting, as well as help from the Holy Spirit.

--Being Watchful, from St. George Greek Orthodox Cathedral in Greenville, SC, USA.

I know that it's not allowed to not focus on anything at all, because the devil will enter the mind in such cases.

Not sure about that, see below.

is there an Orthodox alternative to it, which gives the same or better benefits?

My goodness, yes, the Jesus Prayer: "Lord Jesus Christ, Son of God, have mercy on me, a sinner."

There is also the type of meditation and contemplation done totally in silence, without any words or images or thoughtful activity at all, not even psalmody. The person merely sits in silence, often in the presence of holy icons, and emptying his mind of all thoughts, imaginations and desires, listens to God in silence, the divine “language of the Kingdom of heaven” (Saint Isaac of Syria). This type of meditation, for a person of unceasing prayer, will be the “prayer of silence,” with the “bubbling spring” of the Jesus Prayer as its only foundation and background. In such contemplative prayer and prayerful contemplation, the spirit of man becomes one with the Spirit of God (cf. 1 Cor 6.17).

--The Orthodox Faith by Fr. Thomas Hopko

I think it's fine to focus on your breath, you could even pray "Lord Jesus Christ, Son of God" while breathing in and "have mercy on me, a sinner" while breathing out.

Being Eastern Catholic, I’d say you need to be aware that because of the similarities between some eastern spiritual practices and the practices of the orthodox there is sensitivity amongst the Eastern churches of west’s accusations that we engage in pagan meditation techniques.

I think we need to get over our hang ups with the west’s accusations.

Our traditions; including the Jesus prayer is built around the breath.

As long as you remain theologically sound, practice mindfulness as you will.