"Implementation of the conclusions would undermine the division of competencies between the State and the Entities, seek to give the RSNA veto rights in State level matters, undermine final and binding decisions of the BiH Constitutional Court, a Dayton institution, and determine that the HR’s powers are unconstitutional."
[You can read the veto's legal instrument here].
The country is a two-state federation of the Serb Republika Srpska and the Muslim and Croat Federation of Bosnia and Herzegovina, whose federal government has slowly been strengthened by transferring some of the competences of the states ["Entities"] to the centre, while ensuring that power is shared between the "constituent peoples" fairly at federal level. The process of integration and state-building has been presided over by - and sometimes directed by - the Official High Representative (who is always also the EUSR for BiH) who is empowered by the Peace Implementation Council under the Dayton Agreement. Naturally, any move to weaken the central government and strengthen state power has nationalist implications for each side and threatens the break-up of the state.
Why does this matter for the EU? Well, BiH is practically an EU protectorate. The EUSR is the OHR as well, who can veto legislation and remove elected representatives from office. The EUSR also has a hand in advising the European Commission on BiH's progress and the implementation of Stability and Association agreements and advises the EU on progress in general (visa liberalisation is one of the carrots for BiH's progress). The EUSR is also the focal point for the political co-ordination of the EU's police mission there and EUFOR, as well as helping the HR for CSFP (Javier Solana, who was in BiH in May) on BiH matters. BiH has signed a Stabilisation and Association Agreement and an Interim Agreement, with the eventual goal of EU membership. [The Council's Joint Action that installed the current EUSR, Dr. Valentin Inzko, gives a good picture of the EUSR's remit: link].
All of which means that the EU is heavily invested in the success of BiH as a state. But would the EU be prepared to act in BiH or be able to properly support it if tensions continued to grow to boiling point? The EU finds it hard to act in foreign policy at the best of times, and it will be caught up in a busy, and mainly inward-looking, agenda for the next 6 months: the Lisbon Treaty, a new Commission and Parliament, the Copenhagen Summit on climate change, the financial crisis, etc. Ultimately, of course, the US guarantees stability in BiH, despite European control of the peacekeeping forces and European control of significant political power - still, it would take an almost complete collapse to draw the US in, and the EU would generally be expected to keep the state functioning.
There's no real indication that there's anything beyond an increase in nationalist rhetoric, and Dr. Inzko hasn't mentioned any real instability in his report to the Political and Security Committee of the EU. But it may not take a crisis for EU politics to become entangled in BiH-ian politics: BiH has some ongoing border disputes with Croatia, and these could affect Croatia's accession (since the EU may be unwilling to admit Croatia if it will cause the same problems with BiH-ian accession as Slovenia is with Croat accession). It's likely that there won't be any big (EU-affecting) news coming out of Bosnia and Herzegovina during the Swedish Presidency of the European Council, but it still might be worthwhile keeping an eye on BiH-ian politics given the economic crisis.
It's worth noting that for the next 6 months Carl Bildt, the Swedish Foreign Minister would be a capable pair of hands when it comes to BiH, since he is a former OHR/EUSR himself.