The 30-year-old collapsed in Birdcage Walk near St James's Park, the final stretch of the 26.2 mile course, on Sunday.
Ms Squires was collecting sponsorship for the Samaritans because her mother has volunteered for them for over two decades, and charity spokesperson Rachel Kirby-Rider said: "Obviously we're desperately sad and devastated as an organisation that Claire died whilst running the marathon.
"She was one of 70 runners running for us and our thoughts are with the family at this time and it's really important to us that we continue to support the family during the desperately difficult time for them.
"We're overwhelmed that people are supporting us and the tributes to Claire on her fundraising page and the donations in memory of her are amazing.
"We're a charity that doesn't turnover high sums of money so £500, £1,000 is a huge amount of money to us as an organisation."
As news of the hairdresser's death spread on Monday, tens of thousands of pounds were being donated to her JustGiving page, increasing the total from just £500 the previous day while tributes flooded in on the page and on Twitter.
The website that is hosting the donations has also reportedly agreed to waive their usual fee of 6.5 per cent.
A statement on the official London Marathon website said: "The organisers of the Virgin London Marathon would like to express their sincere condolences to the family and friends of the deceased.
"We would like to emphasise that our immediate concern is for the family of the deceased.
"Our thoughts and deepest sympathies are with them at this difficult time."
Ms Squires' close friend Nicola Short, who raised £1,500 for the Royal Air Force Association with Claire last year in a successful climb of Mount Kilimanjaro, paid tribute to Ms Squires, describing her as an "inspiration" who brought laughter and love to everyone's life.
Tests to establish why Ms Squires collapsed as she neared the finish line are expected to take place over the coming days.
She is the 11th participant to die since the event began in 1981 and the first woman. She died with the finishing line only one bend away, after 25 miles of the marathon.