>Thanks. I did stop and think about that but didn't care enough to look it up. Will try to remember from now on that it's "me." Why is that, anyway?

Well like Mike says but you and John are the object of the sentence - you're what's being seen, so you take the accusative form.
I think this stems from people being drilled from an early age "It's not 'Me and my friend went to the woods' - it's 'My friend and I ...'!" so it sorta gets stuck in the mind that you must use "X and I"

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>>SYNTAX POLICE:
>>... It was also funny to see John and I in complete agreement.
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>>It was also funny to see John and me in complete agreement.
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>>Sorry - bugbear of mine! :-)