I need to travel to Canada from the US ASAP but my US passport is expired. I do have a valid EU passport.
Will I be able to get in and out of the country no problem?
As long as you have proof that you'll be exiting (a bus or plane ticket out of Canada), they should have no problem with you travelling on your EU passport.
Visa and passport requirements: Visas are not required by nationals referred to in the chart for stays of up to six months, except the following:
Visa Required: No, except for EU citizens from Bulgaria, Czech Republic, Lithuania without biometric passport, Poland without biometric passport and Romania. You can stay for up to 6 months.
You will have no trouble entering or leaving Canada on your EU passport (although it may depend on the EU country as to whether you need a visa or not)
Interestingly the place you are most likely to have trouble is returning to the US. US law says "Most U.S. citizens, including dual nationals, must use a U.S. passport to enter and leave the United States.".
I would absolutely make sure you carry a document that proves your US citizenship, otherwise you are going to get trouble at the US border. EU passport holders are generally admitted on visa waiver for tourist visits, but you will be fingerprinted and have to pay a fee.
The other problem (assuming that you are currently resident in the US) is that when you return to the US you are not coming for a tourist visit - you are coming back to live. You will need to carry something that proves you have the right to reside in the US. Otherwise if they see you have a permanent address in the US, or just ask you why you are coming to the US, but you have no documentation to prove you have the right to live there, you can and will be denied entry. I know people that has happened to.