Do not tell the interviewer you took a sickie, FFS
If you are going to, just don't bother turning up because you will NOT be hired.
They are extremely unlikely to ask. If they do, and you don't want to lie, just say "I took a day of leave." Given that "leave" can mean holiday, emergency paid, emergency unpaid, parental leave, sick leave, study leave, jury service or many other things, it's technically true.
I would generally avoid taking sickies full stop but if the job is destroying your health then do it. Given that it sounds like you could cheerfully be signed off by your GP for work-related stress for an indefinite period, it's not even untrue.
References do not ask for specific dates of sickness, just X number of days in Y period (usually 2 years in my experience.) It would be against the law for your current employer to disclose the nature of your illnesses. Unless you are working in a very specialised role (where it's more common usually to use recruitment firms for anyway) they will give a very simple tickbox reference which asks dates of employment, role(s) held, number of sick days, current salary, and would you re-employ.
If you don't get on with your line manager, make sure the reference request goes direct to HR.
I strongly suggest you don't say anything negative about your current employer at interview. If you are asked a question such as "Why do you want to leave?" turn it around and talk about how you want the opportunity to develop your skills in blah blah that their company does, or "I feel that the time has come to work for a larger organisation that will give me exposure to more markets" or "I would like to experience more of a one-on-one relationship with customers that a smaller organisation like yours can foster." Obviously fit your answer to their company.
Good luck!