I just purchased a new ceiling light that uses push connectors rather than bare wires.
My electrical box has two black wires that were originally attached to the previous one black wire of the previous fixture with a wire nut and two white wires that were originally attached to the previous one white wire using a wire not to the old fixture.
How can I connect these wires to the new fixture?
Simply add a pig-tail to the wire nut, then push the other end of the pig-tail into the push connector.
A "pig-tail" is a short section of wire that's added to the existing wire nut. The other end of the wire is stripped bare and is used to connect to "something else". In your case the "something else" will be the push connector of your new fixture.
You don't say were in the world you are, but your mention of "wire nuts" leads me to believe you're in the US. If that's the case, you'll need to use a minimum of 14AWG wire for your short pig-tail (if the light is on a 15A breaker) or, possibly, 12AWG wire (if it's on a 20A breaker). If you're not sure (read the handle on the breaker you turned off before you started this process), your safest bet is to use 12AWG wire, as that's allowed on both a 15A or 20A breaker, but the 14AWG is not allowed on a 20A breaker.
This article* contains this picture which I've updated with some blue arrows to show the "pig-tail" portion of the wiring:
The article does provide a very detailed description of how to wire up a pigtail, so you can get a step-by-step there. Note that in this image, the wire nut on the black wires is pre installation, and not the final installed position. Leaving any bare wire showing (on an insulated wire, not the bare ground wires) is not to code and is asking for a breaker-tripping short circuit.
* This was the first answer in a DuckDuckGo search for "electrical pigtail", no recommendation or endorsement intended or implied.
x amount of #16 in wiring the device to the box and/or decide that #18 is cheaper than #12 so decide to run it through the walls, or...