It's possible, though I've never heard it used that way.

*USUALLY* we are at a whiteboard and we write down the notation or we are sending emails where we also use the notation.

There are occasional cases where someone is talking and I don't know what they're driving at, so I ask them to write it down. I can't think of any examples at the moment, but I'm pretty sure this is not among them. It's extremely infrequent, because usually we're talking in front of the formulae where there is no ambiguity.

I have not seen kallog weigh in on this. I'm curious to know whether he has ever heard or seen it used as
4 meters squared means (4 meters)^2 instead of 4 (meters^2)

Ah...I can think of one case, actually. The term Kilobit means something different to communications engineers and computer designers or system admins. Kb to a comm engineer means 1000 bits; Kb to a system administrator means 1024 bits. I'm not saying there aren't some inconsistencies in terminology between fields, but I have never known the interpretation of "squared" to be among them.