Shengen visas and extended travel

To clarify, there is 1 visa for all Schengen countries as there are not normally any immigrations controls between the Schengen countries, but Americans don't need a visa to enter any of those countries as a tourist. You are eligible to stay in the UK/Ireland for up to 6 months, so you could split your trip between Schengen and non-Schengen countries. As mentioned, US consulates have nothing to with visas for entrance to other countries.
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Bear in mind that as a tourist, you will expected to have enough money in hand - i.e. need to show a bank statement - for housing, food, medical care etc. for all in your family. If the immigrations official at your first port of entry or any entry point between non-Schengen and Schengen countries feels that you don't have sufficient funds, they could refuse you entrance. So make sure to have sufficient funds with bank statements to prove it. Your passport will be stamped or scanned going in/out of Schengen region.

Kate

Simply leaving the Schengen Treaty area and then returning will not "reset" the 90 days, as the limit is worded "90 Days in 180 Days". Not sure how much longer than 90 days you will be traveling or what your plans are, but keep in mind that Great Britain, Ireland, and Croatia are not members, so you have a different 90 days in each of those countries. Also, Switzerland and most of Eastern Europe (Czech Rep., Poland, Romania, Slovenia, and others) are members, but not yet implemented, so they may not be included in the 90 day Visa. Otherwise, rather than the US Embassy in Belgium, Try the Belgium Embassy in the US, they are the ones who would issue an extended Visa.