When would a stock get called away?

<<<Let's say I own a stock trading at $20 and write a covered call with a strike price of $22 and get an option premium of $1. When should I expect the stock to be called away from me; as soon as the stock hits $22 (the strike price) or as soon as it hits $23 (the option buyers' break-even price)?>>>

Neither. Usually the only time it makes financial sense for the option holder to exercise a call option is at expiration. The only exception is when the stock is about to go ex-dividend and the dividend is substantial compared to the amount of time left until expiration. Usually that would mean when the extrinsic value (time value) of the option was less than the amount of the dividend.

Of course it is always possible that a call option on a stock without a dividend is exercised early, but it is a very unusual.

<<<I am curious how near-the-money of a call to sell;>>>

When you sell a covered call (and hold the position until expiration) you are strictly limiting the maximum amount of profit you can make. You will receive the maximum profit if the option is exercised and you shares are "called away." If you would not be happy about selling the stock at the strike price plus the premium you received selling the option, you should pick a higher strike price.

<<<I think the stock will hit the strike price by expiration but not the buyers' break-even... does that make sense to do?>>>

Not really. You have no way of knowing what the buyer's break-even is, and even if you did know you should not care.

Exchange traded options go through a clearing house. When an option holder exercises an option the clearing house makes no effort to match particular holders with particular writers. Even if they did, that would not tell you how much the holder paid for the option.

Consider the following example. Lucy sells a call option to Desi for $2.00 per share. Sometime later the price of the option has gone down to $1.00 per share and Lucy puts in an order to cover (buy to close) her option for $1.00 per share. At the same time you put in an order to sell the covered call for $1.00 per share and it is matched with Lucy's order, so Lucy buys the option from you for $1.00 per share. Since Lucy no longer has any position she is out of the picture. Desi is the option holder and paid $2.00 per share for the option. You are the option writer and received $1.00 per share for the option. The fact that you received $1.00 does not mean that is what Desi or any other option holder paid when buying the option.