Short answer is no. All the REUs do is give you research experience, and this is something you can get through other means, e.g. staying at your institution to work with a professor over the summer, or even over the year.
I wouldn't say an REU is even worth it; lots of professors at top-tier universities have commented that they don't really like or recommend REUs. Or, rather, they are a bit skeptical of their quality as research potential indicators. A lot of them function as businesses, they don't fund international students so unless you're a US citizen you have to shell out the $$. Also, generally speaking, a paper being written on a topic after like one and a half months of research *probably* means it's not a very interesting or hard topic, so that casts doubt on the quality of research. Especially so if it's a paper written by like 4 or 5 people, then that won't help you either because the graduate admissions committee looking at your application won't be able to tell YOUR research potential from a thing written by 5 different people. So in a sense I kinda agree with professors' skepticism of these mass-paper producing REUs.
Some of them are fun though. A lot of the top applicants do them because honestly it's just expected of them. When you're a 4.0 gpa guy from Harvard with grad courses and papers on algebraic topology since freshman year, attending Ken Ono's REU (or another prestigious REU) is just the next thing on the Good Applicant ™ list, but we get those every year, so don't worry about them.
You can get into top 20 schools without any research (even top 10), but I wouldn't recommend it.