Not all companies pay you for how much work you do. For instance, a security guard is not paid per thief he catches, he is paid to be there 8 hours every night so that management can rest easy in the knowledge that there is at least 1 guard present on the premises from 10 pm to 6 am every night. Or a receptionist at the front desk: They are not paid by the number of visitors they help, they are paid by the hour because it is crucial that there always be a receptionist manning the front desk at all times, no matter what, even if nobody shows up.
Software development is often a "paid for work" field where the key thing you provide to the company in exchange for your pay is not availability, but completed work. However, this is not always the case. You can tell by the attitude of the company:
- Do they care if you show up late, leave early, etc?
- What is the attitude to working overtime? Do they pay for it? Do they prefer that you not work outside normal hours?
- When there is concern over someone not pulling their weight, what indicators are looked at? Lines of code or hours of work?
- When a project fails to deliver on time, are you held responsible for the delay?
- Are you often given deadlines, or are there frequent discussion like "by when do you think you can finish this task"?
If they only care about you showing up, forget it. Show up at the usual time and patiently wait while the crappy laptop wastes 15 minutes of company time. Maybe bring a book to read while you do it or make some coffee in the meanwhile.
If management is very strict, work on something else (check your emails on your phone, look at your calendar/agenda) so they can't badger you over it. The company has their own accountants and productivity experts, if they care they will see the inefficiency reflected in their data and independently solve the problem. Maybe they will even ask you if you have suggestions to improve productivity.
Let's say you work 40 hours a week and get $30/hour. Let's also pretend that you spend 20 minutes every day, waiting for the reboots and shut downs, so that you could be spending 15 minutes less if you had a modern computer. That means 3% of your 8 hour day is wasted: The company is paying you $240 for the day and getting 97 units of work, whereas they could be getting 100 units instead if they gave you better equipment. Effectively, they're wasting $7. Is that your problem? Hell no. Your job is to show up at the time the contract says, take the tools the contract says they'll give you, do the work the contract says you will and happily take their money. Budget consulting is extra.
You could indeed show up early, and make sure they get 100 units of work like you think they expect. However, if you do that, you will also be working 8.25 hours a day, not 8. This means that you are now spending 41.25 hours at work and getting $29 for each of them (maybe there's another company that would have hired you for $29.5?). Technically, it's not breach of contract since no one asked you to do it, but doing it essentially amounts to giving your company a donation in the form of unpaid volunteer labor.
Presumably, your contract does not say anything about mandatory unpaid volunteer work (I don't even think it would be legal) so that you are under no obligation to do it. If you know the manager or owner personally, then yes, it's nice to do it, but so would handing him a $100 bill.
On the other hand, if you do get paid for work and not time, it's different: Do whatever it takes to complete your part of the project before the deadline (although this isn't matter of ethics, but practicality: You'll get fired if you don't). If you feel you are forced to spend too much time because of this slow booting, then take it up with management. Even if they are not cooperative, consider just buying your own laptop, and then mentally discounting your salary accordingly when you get competing offers from rival companies.
Lastly, a tangent: This company is doing software development, they hire you at a rate high enough that even a few minutes of your time can pay for a laptop, and they hand you a potato that takes 15 minutes (!!!) to boot? Sounds like management needs to get a clue.