Since as Phil says in his comment on the original question, this is unique and there is no other info on SO on in google, I thought I should also share my results:
In my app FileProvider worked out of the box to share files using the share intent. There was no special configuration or code necessary, beyond that to setup the FileProvider. In my manifest.xml I placed:
<provider
android:name="android.support.v4.content.FileProvider"
android:authorities="com.my.apps.package.files"
android:exported="false"
android:grantUriPermissions="true" >
<meta-data
android:name="android.support.FILE_PROVIDER_PATHS"
android:resource="@xml/my_paths" />
</provider>
In my_paths.xml I have:
<paths xmlns:android="http://schemas.android.com/apk/res/android">
<files-path name="files" path="." />
</paths>
In my code I have:
Intent shareIntent = new Intent();
shareIntent.setAction(Intent.ACTION_SEND);
shareIntent.setType("application/xml");
Uri uri = FileProvider.getUriForFile(this, "com.my.apps.package.files", fileToShare);
shareIntent.putExtra(Intent.EXTRA_STREAM, uri);
startActivity(Intent.createChooser(shareIntent, getResources().getText(R.string.share_file)));
And I am able to share my files store in my apps private storage with apps such as Gmail and google drive without any trouble.