I recently discovered, after a flood in my finished basement of my Condo, that a pipe in the ground under the foundation had collapsed. This was causing any water I used in the house to come back up into the basement. The plumber said it was a collapsed pipe. My insurance agent has told me that they are not responsible for the repairs, because this would be "sewer back up" which requires an additional policy premium and I do not have it. Is a collapsed pipe different from "sewer back up"? I believe it is and this should be covered.

The collapsed pipe in this case is what caused the sewer back up and no, it is not covered. The endorsement that your agent is referring to covers the residual damages to your property as a result of the sewage back up, not the equipment failure which in this case is the collapsed pipe.

In general a homeowner's policy doesn't cover to replace faulty water heaters, toilets, pipes, etc [equipment]- it covers the damage that results from these things breaking down.

So in this specific case of your damage resulting from the backup of sewage, then you would need to have purchased the additional endorsement on your policy. Without the purchase of this particular endorsement, this is considered an exclusion of your policy (read your policy booklet). Most times the endorsement is fairly cheap to add; like $25 per year and often has a separate (lower) deductible than the rest of your policy.

Normally that is not covered and would need the additional endorcements,
Even real "flood" is not covered with 1000's of people in Atlanta are finding out.

On a Massachusetts homeowner policy which includes the endorsement for backup of sewer and drains am I covered for pipes outside my home that are on my property that have been damaged by tree roots?