Change the single quotes to double quotes. And drop the cat (UUoC):
sed "s:foobar:$MYSTRING:" myfile > myfile-individual
From the manpage for bash:
Enclosing characters in single quotes preserves the literal value of
each character within the quotes. A single quote may not occur between
single quotes, even when preceded by a backslash.
Enclosing characters in double quotes preserves the literal value of
all characters within the quotes, with the exception of $, `, \, and,
when history expansion is enabled, !. The characters $ and ` retain
their special meaning within double quotes. The backslash retains its
special meaning only when followed by one of the following characters:
$, `, ", \, or <newline>. A double quote may be quoted within double
quotes by preceding it with a backslash. If enabled, history expansion
will be performed unless an ! appearing in double quotes is escaped
using a backslash. The backslash preceding the ! is not removed.