Well I was hoping your topic was about Armstrong type vinyl flooring that comes in rolls as repairs are fairly easy under the majority of circumstances......
Self adhesive square vinyl flooring looks great when put down and looks that way for a few years but in my expierince with it is once they start to come up and or curl....... it won't be right again.
I haven't tried your plan of action but have gone as far as laying new tiles in the old squares with out long term success. It's kind of like gluing PVC pipe into a PVC fitting,,, most all of the time it sticks but every once in a while they will come apart and no matter how much more glue is added and the two pieces put back togeher, they just won't stay glued. I don't know why but it happens.Your plan of repair may work and is worth a try.
If it were me, I would use the best adhering glue I could find that has a quick dry time and will stick to both surfaces...............or just save my time and pull them all up and roll out some vinyl flooring.
I hope you have success