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In theory, but in programming languages dividing by 0 results in the program saying "I can't do that, it's impossible!" and crashing.
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Depends on the language and context, but mathematics treats division by zero as resulting in 'undefined' (essentially Not A Number) rather than infinity - again, depending on the field and context, but treating it as infinity can produce paradoxes.
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Dividing by zero? Surely it's more of an infinite loop?
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No, for two reasons. One, the button won't be lowered again unless it's a sticky piston.
Two, buttons only produce a second or so of power when pressed. You'd need a lever to toggle between perpetual on/off states.