"On the 9th, in company with ten Elders, I left Independence landing for Kirtland. We started down the river in canoes, and went the first day as far as Fort Osage... Nothing important occurred till the third day, when many of the dangers so common upon the western waters, manifested themselves; and after we had encamped upon the bank of the river, at McIlwaine's Bend, Brother Phelps, in open vision by daylight, saw the destroyer in his most horrible power, ride upon the face of the waters; others heard the noise, but saw not the vision." - History of the Church ~ Joseph Smith, Volume 1, pages 202-03.
The morning after this incident, Joseph Smith received the revelation that later became known as Section 61 of the Doctrine & Covenants. I've included some verses below:
D&C 61: 3-6, 14-16, 18
"3 But verily I say unto you, that it is not needful for this whole company of mine elders to be moving swiftly upon the waters, whilst the inhabitants on either side are perishing in unbelief.
4 Nevertheless, I suffered it that ye might bear record; behold, there are many dangers upon the waters, and more especially hereafter;
5 For I, the Lord, have decreed in mine anger many destructions upon the waters; yea, and especially upon these waters.
6 Nevertheless, all flesh is in mine hand, and he that is faithful among you shall not perish by the waters.
• • •
14 Behold, I, the Lord, in the beginning blessed the waters; but in the last days, by the mouth of my servant John, I cursed the waters.
15 Wherefore, the days will come that no flesh shall be safe upon the waters.
16 And it shall be said in days to come that none is able to go up to the land of Zion upon the waters, but he that is upright in heart.
• • •
18 And now I give unto you a commandment that what I say unto one I say unto all, that you shall forewarn your brethren concerning these waters, that they come not in journeying upon them, lest their faith fail and they are caught in snares; "
Now, the important thing about this that many people miss is that we, as LDS, do not believe the water to be inherently evil or that swimming/bathing is dangerous. Instead, it is directly related to one's obedience to Heavenly Father and the Savior. If you are in the act of rebellion while in the water, you are more susceptible to Satan's power. But that goes pretty much at anytime. When you are disobedient to the commandments of God, you are more likely to be influenced/affected by Satan.
Your uncle probably felt that swimming on Sunday was not keeping the Sabbath day holy. In that respect, he would be right to keep his children out of the water, as he would have felt for them to swim would be sinful and therefore placing them directly in Satan's path. He was not lying to his children, he was trying to protect them.
As to whether or not swimming is breaking the Sabbath, you must decide for yourself. But I will leave with you a quote from Bruce R. McConkie that I think implies that it is breaking the Sabbath to swim that day.
"Sunday being the Lord's Day, it is a day on which men should do the Lord's work, and do it exclusively. There should be no unnecessary work o fa temporal nature, no recreation, no unnecessary travel, no joy riding, and the like. The Sabbath is a day for affirmative spiritual worship, aside from which 'though shalt do none other thing, only let thy food be prepared with singleness of heart.' (D&C 59:13)"