If you take the Regional to Washington and go west on the Capitol Limited, that trains leaves Washington at about 4 PM. That's close enough to dinner time that carryout food purchased at one of the many eateries in Washington Union Station will keep long enough for dinner without requiring a cooler. (I'm not a fan of hauling around a cooler, it's another bulky piece of luggage, and, as has been pointed out, you can't easily get ice refills en route, so after the first day, it becomes useless.)
Breakfast might be a little trickier to bring your own food, as it has to keep overnight, but the
National Cafe Car Menu provides the following: bagel with cream cheese, yoghurt (sweetened), fresh fruit tray, egg cheese and sausage sandwich, muffin and danish, plus coffee, tea, milk and fruit juices. I'd recommend getting breakfast early in the morning to minimize the chance that they'll be running out of whatever it is you really want. On the other hand, I've taken backpacking trips where we just have bagels and cream cheese for breakfast, and the cream cheese is fine even without refrigeration. Hard cheeses also keep very well without refrigeration for a day. In that case, all you'd need the cafe for is coffee or juice.
Also, note that the Capitol is scheduled to arrive in Chicago at 8:45 AM Central Time (9:45 AM eastern), so if the train is on time and you can wait, there are a plenty of places to have a more elaborate breakfast in and around Chicago Union Station.
The California Zephyr leaves Chicago sometime around 2 or 3 PM. As others have said, it does have a dining car that serves coach passengers, but if you want to carry on a dinner, you'll have to have something that can keep a little longer than what you'd buy in Washington. Again, there are lots of places to buy food in and around Chicago Union Station.