Chris ...I don't understand why you would like to stain live cells with H&E stain (usually used for fixed cell/tissues) while you can do so many other/ more productive staining methods like immuno- staining, florescent staining etc.
I don't think Chris's question was if the colorimetric stainings such as H&E could be used in the live cultures without fixation. Chris's question is if that is typically used to stain tissue sections would work on the cell cultures grown in the flasks, chamber slides, coverslips, etc. The short answer yes, it will absolutely work on fixed adherent cells. I must note, however, that since the cellular monolayer can retain much less staining agent, the contrast of this staining may be a bit lower than you might expect. Please see Haematoxylin staining of primary human RPE cells, you can clearly distinguish the cell nuclei but the signal to noise ratio is not as good as you can see when staining tissue sections. That is the reason why H&E is typically known as tissue section staining and far less frequent with cell cultures.