You need to use -webkit-appearance: none; to allow for overriding in mobile browsers.
See this answer: https://stackoverflow.com/a/4812443/737794
Native form elements can be annoying when trying to style cross-browser, so loads of plugins exist that replace the native elements (like spin buttons) with styleable HTML elements. Here's a lightweight plugin that replaces only the spin button elements ->
http://www.egrappler.com/contents/smartspin/demo/spinner.htm
EDIT:
As headkit mentioned, this replaces 'text' inputs, not data-type="spinbox".
Here's the plugin he mentions to replace the spinbox type ->