The spellcheck that marks words spelled incorrectly with a red squiggly line comes from your browser. You'll need to update that dictionary if you want words like "inbox" marked in the good and clear.
Gmail's spellchecker doesn't kick in until you hit the "Check Spelling" link upper-right to the email body. Any words spelt wrong will then be highlighted on a yellow background.
Best Answer
I don't think the feature is present in Inbox, as per this response on Google product forums.
There are other spell checkers you could use though like Grammarly.