echo '"Field1a"|"Field2a"|"Field3a"|"ufghjkrtyrtyfgh$"
"Field1b"|"Field2b
continue on line 2
continue on line 3"|"Field3b"|"ufghjkrtyrtyfgh$"
"Field1c"|"Field2c"|"Field3c"|"ufghjkrtyrtyfgh$"' | sed -nr '/^".*"$/{p;n};:a;/[^"]$|^[^"]/{N;s/(.)\n(.)/\1\2/;ta};p'
"Field1a"|"Field2a"|"Field3a"|"ufghjkrtyrtyfgh$"
"Field1b"|"Field2bcontinue on line 2continue on line 3"|"Field3b"|"ufghjkrtyrtyfgh$"
"Field1c"|"Field2c"|"Field3c"|"ufghjkrtyrtyfgh$"
A sed solution. Every line starting and ending with " will be printed, then 'n' command read the next line and starts a new cycle. If a line doesn't start or end with ", it goes to the loop ':a .... ta', then 'N' command appends the next line, 's' command replaces «lastchar»«newline»«firstchar» (the '(.)\n(.)' part) with «lastchar»«firstchar», then 'ta' command jumps to the ':a' mark only if 's' command actually replaces something (this is the loop). If 's' doesn't replace anything, 'ta' doesn't jump to the mark and sed 'p'rints the resulting line and starts a new cycle with the next line.
The awk solution really seems a lot cleaner. I think my sed solution can be improved.
Edit:
-n option suppress automatic sed output so we print just what we want with 'p'.
-r option is for advanced Regular Expressions.