iPhone: share with instagram

Sorry but you don't really have any control over this. Some sharing panels offer a "Open In..." option in the bottom-right corner. If you check an application that does offer this option (for example Dropbox) you'll likely find Instagram in that list.

For some reason Apple doesn't provide that "Open In..." option in the Camera or Photos apps.

The Facebook and Flickr options are hard-coded by Apple. They always appear in the sharing list even if you don't have the applications installed.

It might be easier to try opening Instagram first and then picking out the picture from your photo gallery. If the picture you're trying to share isn't in your gallery, you can take a screenshot of it in order for it to appear in your gallery. From there you will be able to upload the picture to Instagram. Hope this helps :)

There is no way to share it straight from the photo gallery as of iOS 7. It is much easier to share it straight from the instagram app. You can simply do so by pressing the bright blue camera photo on the bottom of the screen in instagram. You will then be directed to some filters you can choose.
If it still does not work, try quit the instagram app through multitasking and re enter the app.
It might be possible to share straight form the photos app in iOS 8 or you could (not recommended) jail break to get it.

Since Instagram developers do not provide a share extension, this is not possible.

You can, however work around this using the Workflow application using a simple workflow that takes an image and passes it to Post to Instagram action.