Turming unsatisfacory external jscript into satisfactory self hosted jscript

I’m trying to integrate an external search box into my site. However it is not styled how I’d like ( see box 1 in pic ) I’d like to style it like box 2.
I have a site which already has a lovely styled form widget as you can see Here so I was wondering if it is possible to take the external code:

<script type='text/javascript' src='http://www.hotelscombined.com/SearchBox/142080'></script>

and somehow place it on my side and use my styles for it.
Is it possible and how?