John Steinbeck wrote a novel entitled Of Mice and Men. A lot of phrases regarding "mice vs men" are references to this, and "are you a mouse or a man?" may be one of these. I personally cannot recall this phrase appearing in Shakespeare's works, but I am not willing to bet my life on it -- perhaps others more knowledgeable in Shakespeare can answer this part of your question.
Steinbeck got the title for his novel from a poem by Robert Burns entitled To a Mouse, in which the following line appears:
"The best laid schemes o' mice an' men / Gang aft agley."
In this poem, the author has come upon a mouse while plowing the field and this leads him to contemplate and conclude that the plans of mice, just like the plans of men, often come to naught. I believe the "mouse or man" phrase has come into English usage through this route (Robert Burns), but we will see what others think. Don't collect your money yet!
The title of John Steinbeck's novel, Of Mice and Men, is derived from a line in the poem, To a Mouse, on Turning her up in her nest with a plough, written by Scottish poet Robert Burns in 1785. Legend has it that Burns wrote the poem whilst still holding the plough with which he had accidentally destroyed the mouse's winter-nest.
The relevant lines in the poem are:
The best-laid schemes o' mice an' men
Gang aft agley,
Burns is commenting on the fact that even the best made plans of men (meaning humankind) who bestride the world, and in contrast, those of tiny, timid, feeble mice, go awry - suggesting that no one, no matter who or what you are, can ever be quite certain of the outcome of one's endeavors. The poem does not draw a distinct contrast between man and mouse related to strength, power or conviction as the expression, man or mouse does.
The expression was never used by William Shakespeare and cannot be traced to any of his texts. The contrast was first used in 1620 but there is no accurate record of who, or exactly in which context, it was employed. The expression has, however, become popular since then and is now rooted in English vernacular as a somewhat cliched idiomatic expression.
One may assume that the expression's origin is derived from the behavioural mannerisms of mice. They always seem anxious and timid. An observation of their continuously twitching noses and whiskers as well as their uncertain movements and quick scurrying to avoid being noticed, as well as their miniscule size, confirms this notion.