Can I take meclizine HCl 25 mg and Advil (ibuprofen) together?

Yes. There is no interaction caused by taking Meclizine and advil (ibuprofen) together, according to epocrates.
Medication safety. You can use a drug interaction checker whenever you have questions about drug-drug interactions or drug - food interactions. Here is 1 option (Drugs.Com): Your pharmacist is also a good source of info.