Do we say even me or even I?
Do we say even her or even she?
I have broad shoulders, what hair cut would even me out ...(not quite what you meant I think)
Jesus loves even me.
His love reaches even me.
The New York Times stuns even me with its latest betrayal of the nation's trust.
Use 'even I/you/she/he/it/we/they' when you need a nominative pronoun in your sentence. Use 'even me/you/her/him/it/us/them' when you need an accusative pronoun.
Even I should know that! = (Also a person like) I should know that.
It is a strange phrase, an adverb that modifies a personal pronoun. That doesn't happen much!