First something has changed, what is it?. The Squiers' come with .040 - .100 gauge from the factory. Most strings you grab off the store shelf are going to be medium.045- .105: it just the most common gauge so stores stock them. They have more tension than factory. That increases relief. Check what gauge you purchased. My 70's VMJ neck is very easy to tighten when needed; it's a beautiful neck. (Gassing for a black '77 here.)
Second, is this the first time you've changed strings? If so, it's unlikely your rod is shot. More likely the light gauge factory strings just weren't pulling much relief on the neck.
Third. When you tighten the nut, strings to pitch or not, the wood at the nut end anchor and heel anchor can crush as the neck is trying to adjust. "Pre stress" the neck: Grab the head and put the heel against your knee bend the neck back and then tighten the nut. This takes the pressure off the nut anchor and wood: when you release, the nut settles into the wood. The rod can hold your adjustment but it's bad practice to use it to MAKE the adjustment. Loosening it you do not have to worry about "pre stressing". Using this method to tighten the nut, you do not have to loosen your strings. You never need to de-tune to add relief. Just nonsense.
Immigrant says: "Some fuzz be coo', some fuzz be trippin'."