A few days ago a red swollen bump appeared on my penile shaft. It gradually got bigger and seems to have burst overnight. Now there is a slight hole there, and I can see two pieces of something white in there. Is this something I should be worried about?

If this was me, yes, I would have went to see my doctor by now.

Meanwhile, keep it CLEAN with mild soap and water and rinse well! It could be pus in a large pimple/boil/bug bite, but the real problem is that you need to keep bacteria out.
Those of us here who have had cellulitis from infection getting under the skin (in my case, a bug bite that looked like a volcano), can tell you that if we had washed it right away and kept it clean and covered with a antibiotic cream and a bandaid, might not have developed an infection.