The independence of a country can be measured by three things:
1. finance;
2. foreign politics;
3. military affairs.
If these three are controlled by the country, then that country is independent.

Our monetary politics, the finance, economy, trade, etc. are hugely controlled by the EU. EU members have to give up their economic freedom to open up for the other members. The goal of the EU was to create a common market. This is the whole point. Members still have decisions they can make themselves, but there are hard limits on what they can do, and everything has to comply to EU regulations. Plus we can't control what's literally monetary: the Hungarian Forint is pegged to the Euro, so we can't do with it whatever we want (imagine the countries without their own currency but having the Euro instead). By default at the first point our country fails to be independent for this thing.
In the Habsburg empire it was the same, Austria-Hungary was the same in this question. We weren't independent back then, we aren't now.

Our foreign politics also have hard limits, on one hand there is the EU and it's economical dealings with outsiders, but the final boss is actually the NATO and the US on the top. We can't do whatever we want because we have to contemplate those interests first, and only then if there is a leeway.
So the military affairs influence the foreign politics, and the economical ties too. In Austria-Hungary this was the same. We weren't independent back then, we aren't now.

But this is a lesson for the EU too. It aspires to become a sovereign country, but she also has hard limits in foreign politics: it's the same NATO and US which binds it. Just look at what is going on now, US whistles EU jumps. Or that French butthurt when the Australians ditched the submarine deal for the American one, when they could do nothing, called back the ambassadors, then sent it back couple weeks later. Impotence.
So EU needs to take her military affairs in her own hand, if she wants to be the master of its own foreign politics, and be actually an independent state.
Some even wants to abolish the members states, that way creating that supra-state above everyone in here. Which would be still subordinated by American interests