You mention that you do push ups, but do you do any pull-ups or inverse rows?
Typically, it's not uncommon to have Delayed Onset Muscle Soreness (DOMS) due to changing the intensity (weight) or volume (sets * reps). At first glance it sounded like this may be your problem. The DOMS can be quite intense and last for a few days if you really worked things harder than usual.
Your back muscles are involved in push ups, but not to the degree that a proper back exercise uses them. As a result, you might be hitting the rhomboids to a greater degree than the surrounding back muscles so they aren't actively helping out in the process. The way to even things out is to introduce a back exercise.
- Pull ups/Chin ups--one of the gold standard body weight exercises. They hit your lats, traps, biceps, core, etc. Problem is that they are very difficult to do for many people. If you can do at least one, you can quickly add more.
- Inverse Rows--a very accessible and useful back exercise. They are like the opposite of a push up. Your body is straight, just like a pushup, but you are lying on your back and pull yourself up to a bar or table. They'll hit many of the same muscles as the pull ups, but from a different angle. Even if you can't do a single pull up, most people will be able to do this. You can put your feet on a chair to make it more difficult if necessary. Use the same volume as you use on push ups.
Whenever you work one set of muscles, you need to also work the antagonist (opposite) muscles to help your body remain in balance. Failure to do that results in lingering aches and pains, joint inflammation, or muscle spasms. Since you talked about body weight exercise, I provided a couple options that should help that were also body weight exercises.