No, they can't.
You know why? If you don't get the job but the guy who handed over his password does? You talk to the Department for Work & Pensions. It is ILLEGAL to discriminate over the fact that you WOULD NOT break the law. Hence, read my extended threat properly.
You'd probably get more out of them by reporting them than you'd have earned in the first year on the job. This is when the compensation culture / settling out of court works FOR you.
I'm sure the guy who works 100 hours a week, doesn't report missing/broken safety gear, allows his employer to beat him, and gets paid once a year only when they have cash is also "happy" for his job. It doesn't make it any less illegal - or sensible - to hand over personal passwords against legally-binding contracts you have agreed to.
And you have to think what this reflects - this reflects on YOU (I'd happily use this question as a test in an interview for, say, an IT Manager to see whether they will happily breach the DPA just because you asked them to), and on your employer - they obviously don't give a damn about you breaking the law so long as you do what they tell you to.
There are some employers you just DO NOT want to work for. "Just because I need the job" will not excuse any and all evil, no matter the legality. And pulling out the "I need the job" card is the stupidest thing ever. "I worked down a mine that I knew was collapsing because I needed the job"... sorry, no sympathy.
Jobs are important. Money is important. But the courts suing your ass off will hurt you more in the long run, and working for an employer who is NOT BREAKING THE LAW is much more important if you want any kind of job security. Don't let it be you taking the fall, but your employer.
(Hint: I'm currently temping BECAUSE I walked from a former employer who put undue demands on me and had no interest in the legality of actions I was asked to perform / my job / welfare requirements / etc. Since then, it's been revealed that they are doing everything from falsifying disciplinary meeting minutes - for other staff, not me, they knew too well to try it to take it that far as I was on the verge of suing for constructive dismissal anyway - to misreporting finances, to expecting me to casually break the DPA, to wondering why I don't subject myself to physical harm in order to do a job I'm not trained to do. I walked. I got another job. It starts in April. I'll suffer the less-employment in the meantime in order not to work for an employer performing potentially ILLEGAL activities and expecting me - and other employees who have also fled / threatened lawsuits - to collude and/or take the fall when independent auditors come knocking.)