Pull up vinyl floor or leave it down?
It's not clear what the three are proposing. Just removing the vinyl isn't enough, the lauan has to go too. Squeaking has nothing to do with anything here. You don't want the vinyl or the lauan.
Once you're down to the subfloor and it's in good condition, they'll install either a backer board into fresh thin set mortar, etc.
Have any of these fellas confirmed that the subfloor system is good enough for tile installation? That would be the sheets and the framing.
Tile 4 You LLC Troy, MI
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