Your reason to use paragraph (p-tags) for links is completely a mystery to me. You can control styling via CSS, no matter what tag it is. P-tags are for paragraphs, A-tags are for links. That's the most basic thing in HTML semantics I can think of.
The box you are refering to is "-webkit-tap-highlight-color" See http://css-infos.net/property/-webkit-tap-highlight-color