Saying "everything in the universe is connected" is about as useful as saying "everyone has a phone". It is true and completely useless, because the interesting part is always in the details: with whom, how often, over what channel, at what bandwidth.
Over the last twenty years science has given concrete answers to those questions. Let me go through them.
The network of human ties is measurable
Social networks have been studied as graphs since the 1950s, but serious measurement began in the 2000s, when big data arrived. Nicholas Christakis and James Fowler (NEJM, BMJ and other journals, 2007–2009) showed that obesity, smoking, happiness and loneliness spread through a social network up to three degrees of separation: the friends of your friends' friends influence your weight. The effect is small but statistically robust.
How much of this is causal is still debated, but the channels, if they work, are entirely mundane: conversation, shared activities, imitation. No mysticism is needed. The network exists, it works, and it can be measured.
Ecosystems are networks you can see
Robert May (Oxford, Nature, 1972) showed mathematically that an ecosystem that is too densely connected is, on the contrary, less stable: adding links does not by itself add stability. Connectedness can be productive or fragile, and the character of the links matters more than their number.
A tropical forest and a coral reef are both densely connected, yet the forest recovers after a fire while the reef barely recovers after bleaching. The difference lies in how the links are arranged: which nodes are key, where there is slack and where there is none.
The phrase "everything is connected" skips exactly this structure, though it is the only thing that makes systems interesting. Connectedness is a parameter, and it can be tuned.
Quantum entanglement: a connection of a different kind
Entanglement is real (Delft, 2015) and nonlocal, but it cannot be used to send information. That is a proven result, the no-communication theorem.
In practice this means the following. Two entangled particles at opposite ends of the galaxy "know" about each other in a certain sense, but neither Alice nor Bob can use that "connection" to send a message. For that you need a classical channel: light, a signal, a courier.
So citing entanglement as proof that "we are all telepathically linked" misapplies the concept. There is a connection there, just of a very different kind from the one people want.
What remains
Three layers of connectedness actually work.
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Social. Behaviour, health and mood spread through real contacts up to three degrees out. This is measurable and useful in practice: it pays to choose your surroundings, because they filter what reaches you.
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Ecological. Ecosystems are linked by food chains, chemical cycles and migrations. Knock out one node and a cascade runs through the network, most visibly when keystone species go extinct.
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Informational and physical. If gravity is emergent (Verlinde) and the universe is a neural network (Vanchurin), then connectedness at the deepest level simply is the architecture of the universe. That is speculation, but testable speculation.
"Everything is connected" in the sense of "there is a path between any two points" is true but trivial. "Everything is connected" in the sense of "there is bandwidth between them and something flows" is far more interesting, provided you leave the magic at the door and count the graph.