Did I remove the tick the right way?

I have a pure breed German Shepherd who is very active and loves being outside, and ticks are an inevitable consequence that comes with it. To start with the tick you saw is probably a female dog tick and her color was most likely darker because she was filled with blood. Pulling these kind of ticks is the best thing to do. However, as you might have already realized, DO NOT simply pull them from the swollen white/grey part because that will just burst. Hold/pinch them from the base of their head (the place from where it's stuck on the dog) and instead of pulling it out at right-angles slowly scrape it up at a slightly lower angle. This will make it easier for you and hurt the dog less. Once it's out immediately kill it by either crushing it or putting it in boiling water. Trust me if you leave it, it will be back.
Next if you found a female there's probably going to be a male nearby which looks like this:

so look for that. These types of ticks love a dirty unwashed dog more than anything. We wash our dog almost every other day and wash her feet after every time she goes on a walk. So wash your dog at least 3 times a week in the tick season. Use a soap which is a tick prevention / treatment shampoo instead of the regular dog soap. You can also try using tick repellent powder (although be warned They have a very strong smell). Something I've found very effective is using apple cider vinegar mix as a tick repellent. Just add two parts apple cider vinegar to one part water in a spray bottle. You can also add a few drops of peppermint oil if you like. Spray this mix all over your dog in the morning and evening (or when your dog goes out). 1-2 sprays on the back and 1 spray on the neck, each of the feet and the end of the tail should do the trick. (just make sure while you're spraying it doesn't enter the eyes or nose of your dog) Finally get one of those really thin "tick removing" combs which look like this.

Groom your dog with this daily at least once. This is one of the most effective ways to remove ticks without using any chemical treatments.

And regarding your question about eggs, although I'm not an expert, I don't believe that's the case for your dog because I get the impression that you don't find ticks that often on your dog. Besides, from what I know, usually these ticks would lay eggs in open wounds and/or extremely unclean dogs, but again I don't thinks that's the case for you. Nevertheless, you should always take precautions and I hope some of the ways I've mentioned above will help you do that.