No, it would be "the birds protected their nest"...
Or,
"The birds' protection of their nest is a sight to behold."
Actually "the birds protected their nest and "The birds' protection of their nest is a sight to behold." is a bit off the mark.
Read her post and check for the plurals; more than one bird more than one nest.
Her grammar is correct as well as syntax; she is presenting the protected state of more than one bird's nest; "the birds' nests..."
You could alternately present a similar statement the way you have indicated, but to match Harritte's intent you would need to add an "s" to "nest" in your statement(s) to represent the existence of multiple nests. "the birds protected their nest
s"...
I am now done being a grammar and spelling nerd...