Apparently, the popularity of the video (the friends even appeared on Ellen recently) led the two teens to file federal trademark applications for the phrase DAMN DANIEL in the United States Patent and Trademark Office on February 24th (with the help of attorneys at Varner & Brandt, LLP).
The teens filed one application in Class 041 for "[e]ntertainment in the nature of viral video media, namely, online videos broadcast through social media networks" and the other in Class 025 for "[c]lothing items, namely, t-shirts, hats and shoes." The application in Class 025 was filed on an intent-to-use basis (meaning the teens are not using the trademark on those goods yet but have a bona fide intention to do so soon) but the application in Class 041 is based on actual use in commerce, with a first use date of February 15, 2016 (the date Josh posted the video on Twitter).
The specimen submitted with the application in Class 041 is seen above.
If successful in obtaining these federal trademark registrations, the teens will obtain the exclusive, nationwide right to use the term DAMN DANIEL on goods or services related to those listed on their applications. Whether those rights are enforced by the teens is yet to be seen. Given the viral nature of the video, I have a feeling enforcing them will not be easy.