I came across these capacitors in a kit and was wondering what type they are so I can buy more for use in button debouncing circuits. Unless I'm reading them wrong, they are 220 nF and 100 nF respectively.
You have the base values correct. They also have a ±10% value tolerance (K) and a maximum working voltage of 63V.
For debouncing you can use physically smaller capacitors than those by going to ceramic types. As Ignacio says, the ones you have are film capacitors 100nF and 220nF +/-10% 63V working voltage.
From the size and application, they're polyester film. The construction is sometimes called "box" type (because of the molded box rather than dipped construction) and the lead pitch is 5.0 mm (from the breadboard, or it could be 5.08mm).
Here is a similar Kemet part for 100nF.
Here is an electrically similar ceramic 0.1uF/50V 10% capacitor. Since your logic voltage won't exceed 5V normally, anything with 10V, 16V or higher voltage rating is fine.
If you were to lay out a PCB, an equivalent SMT ceramic capacitor would be very tiny and cheap. Such as this one from Samsung (100nF/10V 10% X7R), which is 0.86 USD for 100 pieces.