The EU recently published a series of technical notes on Brexit preparedness.
Here are a few extracts from the “Dot EU” technical note (EU Brexit preparedness notice on the same topic).
“As of the withdrawal date, undertakings and organisations that are established in the United Kingdom but not in the EU and natural persons who reside in the United Kingdom will no longer be eligible to register .eu domain names or, if they are .eu registrants, to renew .eu domain names registered before the withdrawal date.” 
“Accredited .eu Registrars will not be entitled to process any request for the registration of or for renewing registrations of .eu domain names by those undertakings, organisations and persons.” 
“Where, as of the withdrawal date and as a result of the withdrawal of the United Kingdom, a holder of a domain name does no longer fulfil the general eligibility criteria pursuant to Article 4(2)(b) of Regulation (EC) 733/2002, the Registry for .eu will be entitled to revoke such domain name on its own initiative and without submitting the dispute to any extrajudicial settlement of conflicts in accordance with point (b) of Article 20, first subparagraph, of Commission Regulation (EC) No 874/2004.” 
Clearly this is subject to the negotiations currently underway. It is worth making plans if you make use of this TLD as soon as possible, to ensure uninterrupted Internet services.
Reference 1: Notice to stakeholders – withdrawal of the United Kingdom and EU rules on .EU domain names, European Commission, 28 March, 2018 https://ec.europa.eu/info/sites/info/files/notice_to_stakeholders_brexit_eu_domain_names.pdf