Remove parquet flooring adhesive by using a heat gun

Are the tongues necessary? Can I just take my circular saw right between the boards, cut them way, and then screw them back down afterward?

You've been misinformed. The point of tongue-in-groove planking is to keep the floor boards from twisting, slipping and sliding against each other and squeaking (or squeaking more in your case). You still need to secure a tongue-in-groove floor to the subfloor or joists.

Joists almost never squeak on their own. Squeaking comes from two pieces of wood rubbing against each other - usually because a floor hasn't been adequately secured.

Now if you DO need to cut away the flooring, then you should cut along the seams, and then buy Hardwood Floor Spline and rout out the old tongue so you end up with two grooves facing each other, and use the spline to rejoin the flooring when you reattach it.

But BEFORE you do that, I'd get a bunch of 2" coated screws and drop them through the subfloor into the joists below and see if that fixes the squeak. It's subfloor so you don't care about the face, and if it doesn't work, they're easily removed.

*Tip - use a hammerdrill when putting in the screws

One comment on the advice you read is that I don't see a thing in there about deflection - evidently Mike Holmes is sure that if he merely has 1-1/4" of subfloor, all is good. Real tile specs tend to involve a concept called deflection, and that's the distance part of the floor system moves when loaded, relative to the span you measure over. L/360 is a typical spec - L/180 is twice as bad. (It's usually a lowercase L, but on the computer I'm using a capital to be clear that it's an L.) I happen to have a 3/4" subfloor that's L/587, so it really doesn't need another half-inch on that basis.

Consider what a tile manufacturer has to say... (I'm not going to cut and paste their extensive text here, or rewrite it all) but for one thing they specifically mention setting the backer board in a mortar bed, not "gluing" it down. I'd also venture to sneak in that if you really want "the best" (and will you pay for it these days) look at the "old fashoned mortar bed" (TCNA (Method F145-02)) mentioned at the end of their article. That, of course, means you need to establish an adequately rigid base 3/4" or 1-1/4" + tile thickness below the surrounding floor level, which is why it's not popular these days. But it was a VERY good method, as a lot of old houses demonstrate.

Kinsman Tile