filters are made for stuff like this. In your repo,
cat >.git/info/saved-connection <<EOD
<setting name="ConnectionString"><value>Server=prodServer;Database=myDataBase;</value></setting>
EOD
cat >.git/info/my-connection <<EOD
<setting name="ConnectionString"><value>Server=myprivateserver;Database=myDataBase;</value></setting>
EOD
git config filter.use-my-connection.smudge 'sed -f ".git/info/use-my-connection.smudge"'
git config filter.use-my-connection.clean 'sed -f ".git/info/use-my-connection.clean"'
cat >.git/info/use-my-connection.smudge <<EOD
/^<setting name="ConnectionString">/ {
w .git/info/saved-connection
r .git/info/my-connection
d
}
EOD
cat >.git/info/use-my-connection.clean <<EOD
/^<setting name="ConnectionString">/ {
w .git/info/my-connection
r .git/info/saved-connection
d
}
EOD
echo >> .git/info/attributes path/to/app.config filter=use-my-connection