Is it sinful to be a "prayer squirrel"?

Sin is the transgression of God's commandments so, no, unlikely.

Leviticus 4:1-2 ​ And the LORD spake unto Moses, saying, Speak unto the children of Israel, saying, If a soul shall sin through ignorance against any of the commandments of the LORD concerning things which ought not to be done, and shall do against any of them:

Leviticus 4:27 (KJV) And if any one of the common people sin through ignorance, while he doeth somewhat against any of the commandments of the LORD concerning things which ought not to be done, and be guilty;

Romans 5:13 (KJV) (For until the law sin was in the world: but sin is not imputed when there is no law.

There is no law against being a prayer squirrel thus it is not a sin.

Sin requires assent of the will. Therefore, an involuntary distraction is not a sin.

Actions that increase the likelihood of distraction, however, may be a sin. For instance, one might have to seek out a private place not only to avoid notice, but to avoid noises that distract. A false spirituality that refuses to admit that you are distracted by something else should be fought. Even if the distracting thing is, in fact, not so important as what you should be praying about, you should, at the very least, pray for aid to deal with the distraction. As in Screwtape Letters, where Screwtape points out that distracting the patient at his prayers with thoughts of his girl was an utter failure if it merely led him to pray for help against distraction.

Thankfully, you gave an example of what being "a prayer squirrel" means. What remains outstanding is a clear understanding of what it is to sin. Some answers have claimed that one's will must be involved in committing a sin - that a sort of decision has been made to do something wrong in God's eyes, or that one has chosen to lapse rather than to resist a temptation to do wrong. Or that sin is a transgression of God's commandments. Ah, well, there are a couple of problems with viewing sin that way.

First, it does not take into account sins of omission. It only deals with sins of commission. Yet the Bible tells Christians that if they fail to do something good, they have sinned (e.g. if you have two coats and see someone without even one, you should act.) See also Jesus' parable of the good Samaritan.

Second, There is no command against being detracted whilst in prayer. We have poor Eutychus who fell out of an open window and died, having falling asleep during a lengthy discourse of Paul, but Paul raised him back to life! (No mention of whether a long prayer was going on then...) Eutychus succumbed to physical tiredness, not to sin.

Then there's the case of the disciples falling asleep at night while Jesus was praying in the garden of Gethsemane. Were they accused of sin? It distressed Jesus, and they felt ashamed, but even though we may well consider them to have sinned (failing to keep Jesus' command to them), their physical tiredness was the problem, not any exercise of the will. "The spirit is willing, but the flesh is weak" (Matthew 26:41). But if anyone wants to consider weakness of the flesh as sin, I would not disagree. I think that is true.

So, let me put my hand up immediately to admit that due to weakness of my flesh, I am sometimes guilty of having my mind wander while in prayer (either listening to others' prayers, or saying my own private ones). It is perfectly possible to sin in holy matters, and prayer is a most holy privilege and duty. I have to confess that to God and seek forgiveness when it happens (for it is a case of 'when ', and not 'if'.) Yet it remains a comfort to me that saints of old have turned to God in prayer when unable to sleep, and who knows but that their prayers led them into sleep? Could there be a better way to transit from wakefulness to sleep than whilst in prayer? That way, one's last waking thought was on God and (when one prays the minute they are awake) their very first wakening thought is on God.

None of us can judge another on this private matter of prayer. Colossians 4:2 says, "Continue in prayer, and watch in the same with thanksgiving." A.V. So, thank you for raising this question, which admonishes me to be more diligent, so that I sin not in holy matters.

Is it sinful to get distracted while praying?

While the Bible admonishes believers on our prayer lives in many places, to pray often (1Th 5:16-18), to seek the LORD in all we do (Pro 3:5-6), and to devote ourselves to lives of prayer (Col 4:2), the gift of prayer itself is never seen as an issue related to sin or not. In other words, nowhere in the Bible are people condemned for their lack of prayerfulness.

Not praying is not an issue of sin or sinfulness, rather it is an indication of the health of our relationship with God - if we are close to God, we will feel the need to be in prayer, if we are far away from Him, we won't.

So is it a sin to get distracted while praying? Hermenuetically, the verse in question isn't really saying that; instead it is encouraging people not to be so distracted that they do not take time to pray. Be vigilant in your prayer life! The idea that somehow being distracted during prayer time could be seen as a sin seems overly legalistic and too quick to condemn, that is not at all what the apostle Paul is communicating in the greater chapter in question.

If you are feeling distracted during prayer time, try to practice mindfulness techniques (focusing on the moment, letting go of thoughts that come into your mind, and refocusing on your conversation with God). The more time you spend consistently in times of prayer, the easier and better your prayer times will become.