Can my American children re-enter the USA by International flight with a passport card, as their passport book has expired?

The problem has two layers.

Being allowed into the country — they are US citizens. They cannot be refused entry into the US. However, immigration can detain them as long as is necessary to confirm their citizenship. The passport card should cover this task, however you are technically are violating the law (as phoog discusses at length). Expect considerable delay and a stern lecture, finger-wag and possible fine.

Being allowed to board an airplane (or boarding pass in preclearance airports) — the bigger problem is the airline may be fidgety about allowing you on board the plane, if your credentials are not in order. And unlike CBP, they are not obliged to work with you, and they do not have all of CBP’s resources to confirm your citizenship. They also face serious financial penalties for not pre-checking your credentials. As such, they are likely to refuse to let you get on the airplane. This will get decided ad-hoc at the gate, without any sort of appeals process, and when you miss the plane, you will eat the cost of the flight.

The consulate here is not giving me a solid reason why I need a more costly passport book.

The relevant federal regulation is found at 8 CFR 235.1. It starts by saying that everyone needs a passport unless they qualify for one of several exceptions, found both in that part and at 22 CFR 53.2. In addition to the exception for a passport card, there are exceptions applying to children under 19 or 16, depending upon the circumstances. However, all of these exceptions apply only for entry at a land or sea port of entry. They do not apply to international air travel.

The passport card exception is found at 8 CFR 235.1(b)(1):

Passport card. A U.S. citizen who possesses a valid unexpired United States passport card, as defined in 22 CFR 53.1, may present the passport card when entering the United States from contiguous territory or adjacent islands at land or sea ports-of-entry.

The exceptions for children are found at 8 CFR 235.1(b)(8):

(8) Children. A child who is a United States citizen entering the United States from contiguous territory at a sea or land ports-of-entry may present certain other documents, if the arrival falls under subsection (i) or (ii).

(i) Children under Age 16. A U.S. citizen who is under the age of 16 is permitted to present either an original or a copy of his or her birth certificate, a Consular Report of Birth Abroad issued by the Department of State, or a Certificate of Naturalization issued by U.S. Citizenship and Immigration Services when entering the United States from contiguous territory at land or sea ports-of-entry.

(ii) Groups of Children under Age 19. A U.S. citizen, who is under age 19 and is traveling with a public or private school group, religious group, social or cultural organization, or team associated with a youth sport organization is permitted to present either an original or a copy of his or her birth certificate, a Consular Report of Birth Abroad issued by the Department of State, or a Certificate of Naturalization issued by U.S. Citizenship and Immigration Services when arriving from contiguous territory at land or sea ports-of-entry […]

The provisions at 22 CFR 53.2(b)(11) are essentially the same:

(11) When the U.S. citizen is a child under the age of 19 arriving from contiguous territory in the following circumstances:

(i) Children under age 16. A United States citizen who is under the age of 16 is permitted to present either an original or a copy of his or her birth certificate, a Consular Report of Birth Abroad, or a Certificate of Naturalization issued by U.S. Citizenship and Immigration Services when entering the United States from contiguous territory at land or sea ports-of-entry; or

(ii) Groups of children under age 19. A U.S. citizen who is under age 19 and who is traveling with a public or private school group, religious group, social or cultural organization, or team associated with a youth sport organization may present either an original or a copy of his or her birth certificate, a Consular Report of Birth Abroad, or a Certificate of Naturalization issued by U.S. Citizenship and Immigration Services when arriving in the United States from contiguous territory at all land or sea ports of entry…

Now a US citizen cannot be denied entry to the US if the US citizen can get to the border and prove his or her US citizenship by any means. The problem here is that the airline is not bound to transport US citizens based on any proof of citizenship; they are required to demand identification that meets the regulatory requirements.

Yes, they are correct. The passport card was designed for land and sea ports of entry and is not allowed for international air travel.

Passport card page from the State Department