Holiday /Work in EU

Yes.

Unless you get a working visa of some sort.

Unsure if you are even permitted to "work" in the EU without a work visa after 1.1.21 - so I assume you will be "working" for a UK legal entity and visiting the EU for "meetings".

Counting for days begin at one second after midnight on the day you arrive in the Schengen area so if you landed at say 23:59 that would already be that whole day gone.

In that circumstance you might think about dragging your feet on the way to immigration so your 'arrival' rolled over to the next day! 

It ends at midnight on the 90th day so as late a flight or ferry out as possible would maximise you stay. 

Within the 90 days you will be free to spend any combination of days in any country but the overall total of 90 stands.

I don't know what sort of work it is you do but I think unless it's for an international conglomerate with presences in each of the countries in which you envisage working, and are officially posted to, then outside of that it's difficult to see how you could continue with the sort of itinerate work regime you allude to. 

At very best it would be a bureaucratic nightmare as you'd need work visas for each country and those would have to be applied for and granted before you entered them. 

To obtain a work visa you will need to be in possession of a job offer and in turn the prospective employer will need to demonstrate that the vacancy you are being recruited to fill cannot be filled by one of their own citizens.

In the past many employers have been willing to turn a blind eye to the niceties of employing citizens from other EU member states but illegally employing non EU 3rd country citizens is a whole different scenario with potential penalties I think few will be willing to expose themselves to. 

I used to work for a UK-US company. The UK company (my employer) was a separate legal entity from the US company, but both companies were subsidiaries of a holding company that was traded on both stock exchanges. But, at the operating level, the companies acted as one.

I sometimes spent around a third of my time in the US.

To avoid visa (and personal taxation) issues, I worked in the US in a consulting capacity, and never for more than a couple of weeks at a time. 

When interrogated by US Immigration, I had to be circumspect in my "prepared answers". I could never refer to accountability for staff, budget or assets in the US.

This might be how it works with the EU going forward. 

But clearly, if you have a role with staff, budget or asset accountability, reporting into/salary paid by legal entities registered in EU countries, one or more work visas may be required.  

Interesting, worrying, inevitable

https://www.theguardian.com/business/2020/dec/10/uk-business-travellers-to-eu-face-fines-over-post-brexit-permits?CMP=Share_iOSApp_Other&fbclid=IwAR2AIIhvC5acTP_INtFPHau-3WkposNg4R1WCkFbVh9UQOYTkumGa2JrdHY