By land: Likely successful but not guaranteed
U.S. citizens and lawful permanent residents, regardless of age, are technically exempt from passport requirements under s. 52(2) of the Canadian
Immigration and Refugee Protection Regulations. Legally, there is no passport requirement for U.S. citizens.
However, you must satisfy to the immigration officer that you are in fact a U.S. citizen, and only a valid U.S. passport or NEXUS card or enhanced Driver's Licence issued to a U.S. citizen is definitive proof of that for Canada. With this in mind, Canadian government has decided to not publicize the fact anymore and generally recommend to always have a passport, even in absolute wordings that may be in fact contrary to what the law says. Unlike for Canadian citizens and permanent residents, the Canadian government is generally under no obligation to help a foreign national to establish identity and citizenship for purpose of entry.
CBSA routinely admits U.S. citizens without a valid passport (recently expired passport is generally not a problem), sometimes even on oral declaration alone; however, they also routinely deny entry to people without a valid passport. For children, U.S. birth certificates is usually acceptable to CBSA as proof of citizenship. It is an exercise of discretion, children are generally considered low-risk when travelling with their families, but other factors (e.g. whether the child is travelling with family with the same last name, whether the child looks older or younger, the supposed purpose of travel) may cause the officer to exercise greater scrutiny (e.g. more questions on the identity, requesting other documents, and potentially denying entry).
By air: Unlikely
While the law governing the admission of foreign nationals is mostly the same for land and air border, air travel has an additional layer of regulation and document verification. Air travel is subject to security regulations, which for Canada include passport (or NEXUS card) requirement for any one looking 18 years of age or older. Additionally, the airlines will also check the documents to avoid being responsible for the removal costs if Canada denies entry. You are likely to be denied boarding with birth certificate alone.